when ,
but means something else when
(and goes unused altogether for other values).
this is a doggone shame. surely we oughta have or something to denote inverses.

the rest don’t bother me.
as for and you might as well complain
that it’s easy for beginners to confuse d with b
(this is presumably why earlier generations were taught
to “mind their p’s and q’s”; i expect this is passing
from the language pretty quickly though …).
and should be rare-to-nonexistent
in our context (“precalculus”, say).
that things not known to commute
must not be assumed to commute
isn’t something we bring on ourselves at all;
it’s a fact of nature. if students find it confusing,
well, that’s job security for us. likewise for the
contravariant behavior associated, as in jonathan’s
example, with adding before applying a function.

oh, and “onto”. why not get into it?
my opinion? — thanks for asking.
the real problem is with
the usually-ill-defined term “range”
which should be split forthwith
into “target” and “image”.
thus, given that satifies, we can say that has the full set of real numbers as its “target”
but if moreover, we know that ,
then has as its “image”
(and the question of “onto” simply doesn’t arise).

(Whoops could you please remove the comment above! I am not with it today. :o) [Done! I fixed the LaTex too. JB]

I have never got confused with f(g(x)), but I think confusion about arises, because I seem to recall something about dy/dx= . (Or something similar?!)

Vlorbok: About the onto business – I definitely agree! I will re-read what you wrote (whilst I am fully alert) because something tells me it will make sense. Because of this onto business, I hate surjective functions and the word onto. (Memories of my linear algebra exams and kernels spring to mind). GAH. (Finally a place where I can express my dislike for them!)

Vlorbik is even more right because the distinction between “target” and “image” is what real mathematicians do. In fact, when you generalize away from “functions” between “sets”, everything has a target, but we can’t always even define what an image is.

On the other hand, there’s a big schism between your seemingly more sensible definition of composition and the more traditional way. The usual way says that means “first do , then do to get “.

Oh, I didn’t mean you’d say it in those terms. Still, it’s just as easy to say “target” and “image” instead of “range”, and closer to what real mathematicians say. And if the kids do eventually get to that level, so much the better.

Also, I agree that it’s just as easy to say “target” and “image”. I’ll be bringing it in to our conversations – but probably not the week before finals (I don’t want to confuse them). This will have to wait until second semester.

It’s really about 2/3-1/3, in my experience. John Baez makes specific note that he breaks with the standard in his quantum gravity lecture notes to write it your way, as well as in most of his higher-dimensional algebra papers. Most abstract algebra textbooks I’ve seen write with the convention I stated, though the Wikipedia entry on function composition does it your way.

Really it doesn’t matter, as long as whoever’s talking or writing is clear about which order of composition is intended. It also modifies things like whether operations are covariant or contravariant, and it swaps the definitions of left and right modules over a ring. If only mathematics had a body like l’Académie française to just settle the question and impose a standard.